Name: finding absolute extrema on a closed interval
Uploaded: 2026-01-12T12:24:20.000Z
Duration: 1 min 21 s
Channel: JoVE

Déterminer où une fonction atteint ses valeurs les plus élevées ou les plus basses aide à décrire son comportement global. Identifier ces valeurs, c’est comme trouver des sommets et des vallées le long d’un sentier de randonnée.Une fonction peut atteindre ces valeurs aux nombres où sa dérivée est nulle ou indéfinie. Ces nombres sont appelés nombres critiques — nombres dans le domaine où la dérivée est nulle ou indéfinie.Selon le théorème de Fermat, si une fonction a un maximum ou un minimum local en un nombre et que la dérivée y existe, alors la dérivée doit être nulle.Cela signifie que les extrêmes locaux apparaissent toujours aux nombres critiques, bien que tous les nombres critiques ne correspondent pas à un extrémium.Certaines fonctions présentent des coins vifs où la dérivée n’est pas définie, mais un extremum local peut tout de même être présent.Pour trouver les valeurs absolues les plus élevées ou les plus basses sur un intervalle fermé, on utilise la méthode des intervalles fermés. Cela implique trois étapes : identifier les nombres critiques dans l’intervalle ouvert, évaluer la fonction à ces nombres, ainsi qu’aux extrémités. Les plus grands et les plus petits de ces résultats donnent le maximum absolu et le minimum.

nombres critiques et méthode de l’intervalle fermé

Critical Numbers and the Closed Interval Method

Comprendre les valeurs maximales et minimales d&#x2019;une fonction est essentiel pour analyser son comportement global. Ces valeurs, souvent appel&#xE9;es extr&#xE9;mums, permettent de mieux saisir la mani&#xE8;re dont une fonction se comporte sur son domaine. D&#x2019;un point de vue math&#xE9;matique, les extr&#xE9;mums peuvent &#xEA;tre locaux &#x2014; repr&#xE9;sentant des maxima et des minima dans une r&#xE9;gion limit&#xE9;e &#x2014; ou absolus, indiquant les valeurs les plus &#xE9;lev&#xE9;es ou les plus basses sur l&#x2019;ensemble d&#x2019;un intervalle.
Les extr&#xE9;mums d&#x2019;une fonction se produisent aux nombres critiques, c&#x2019;est-&#xE0;-dire en des valeurs du domaine pour lesquelles la d&#xE9;riv&#xE9;e est soit nulle, soit non d&#xE9;finie. Selon le th&#xE9;or&#xE8;me de Fermat, si une fonction admet un extr&#xE9;mum local en un point et qu&#x2019;elle est d&#xE9;rivable en ce point, alors sa d&#xE9;riv&#xE9;e y est n&#xE9;cessairement nulle. Toutefois, la r&#xE9;ciproque n&#x2019;est pas toujours vraie : tous les nombres critiques ne correspondent pas n&#xE9;cessairement &#xE0; des extr&#xE9;mums locaux. Certains peuvent signaler des points d&#x2019;inflexion ou des zones o&#xF9; la pente est nulle, sans caract&#xE8;re extr&#xE9;mal.
Il convient &#xE9;galement de souligner que les nombres critiques incluent les points o&#xF9; la d&#xE9;riv&#xE9;e n&#x2019;est pas d&#xE9;finie. Ces situations apparaissent fr&#xE9;quemment pour des fonctions pr&#xE9;sentant des angles vifs ou des points de rebroussement, comme la fonction valeur absolue, pour lesquelles la diff&#xE9;rentiabilit&#xE9; classique fait d&#xE9;faut, alors qu&#x2019;un maximum ou un minimum local peut n&#xE9;anmoins exister.
Pour d&#xE9;terminer les extr&#xE9;mums absolus d&#x2019;une fonction sur un intervalle ferm&#xE9; [a, b], on applique la m&#xE9;thode de l&#x2019;intervalle ferm&#xE9;. Cette d&#xE9;marche repose sur trois &#xE9;tapes :
<ul>
<li>Identifier tous les nombres critiques dans l&#x2019;intervalle ouvert (a, b)</li>
<li>&#xC9;valuer la fonction en ces nombres critiques</li>
<li>&#xC9;valuer la fonction aux extr&#xE9;mit&#xE9;s a et b</li>
</ul>
En comparant l&#x2019;ensemble de ces valeurs, la plus grande et la plus petite sont respectivement identifi&#xE9;es comme le maximum absolu et le minimum absolu sur l&#x2019;intervalle. Cette approche syst&#xE9;matique garantit que tous les points susceptibles de correspondre &#xE0; un extr&#xE9;mum sont pris en consid&#xE9;ration, offrant ainsi une analyse compl&#xE8;te du comportement extr&#xE9;mal de la fonction sur un domaine donn&#xE9;.

Nombres critiques et méthode de l’intervalle fermé

Comprendre les valeurs maximales et minimales d&#x2019;une fonction est essentiel pour analyser son comportement global. Ces valeurs, souvent ...

critical-numbers-and-the-closed-interval-method

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Calculus

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Applications of Differentiation

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KEY TERMS AND CONCEPTS

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finding absolute extrema on a closed interval

Absolute extrema on a closed interval are found by checking critical numbers and the endpoints. A function’s maximum and minimum values help describe its overall behavior across its domain. These values are called extrema, and they can be local or absolute.

Local extrema are peaks and valleys in a limited region. Absolute extrema are the highest or lowest points on the entire interval. Critical numbers are the domain values where the derivative is zero or does not exist. According to Fermat’s Theorem, if a function has a local extremum at a point where it is differentiable, then the derivative at that point must be zero.

A critical number does not always mean there is a local extremum. Some critical numbers mark inflection points or flat regions instead. Critical numbers also include points where the derivative is undefined. This can happen at sharp corners or cusps, such as on an absolute value function, where a local maximum or minimum may still occur.

The Closed Interval Method is used to find absolute extrema on an interval written as [a, b]. First, identify all critical numbers in the open interval (a, b). Next, evaluate the function at those critical numbers. Then evaluate the function at the endpoints a and b.

After all of these values are found, compare them. The greatest value is the absolute maximum, and the least value is the absolute minimum on the interval. This method makes sure every possible extremal point is checked, giving a complete picture of the function’s behavior on a closed interval.

Finding Absolute Extrema on a Closed Interval

finding-absolute-extrema-on-a-closed-interval

absolute and local extreme values

Finding Max and Min on Function Graphs

Absolute and local extreme values show the highest and lowest points on a function graph. Absolute maximum and absolute minimum values describe the highest and lowest values across the entire domain. Local maximum and local minimum values describe the highest and lowest values within a smaller part of the graph. Periodic functions such as sine and cosine can have extreme values at many points. Their repeating pattern creates multiple local extrema. In these cases, the derivative becomes zero...

critical numbers and the closed interval method

Absolute extrema on a closed interval are found by checking critical numbers and the endpoints. A function’s maximum and minimum values help describe its overall behavior across its domain. These values are called extrema, and they can be local or absolute. Local extrema are peaks and valleys in a limited region. Absolute extrema are the highest or lowest points on the entire interval. Critical numbers are the domain values where the derivative is zero or does not exist. According to Fermat’s...

rolle’s theorem

Rolle’s Theorem and Zero Slope Points

Rolle’s Theorem explains when a differentiable function must have a flat tangent inside an interval. It applies to a real-valued function on a closed interval, which means the endpoints are included. The function must also be continuous on the whole interval and differentiable on the open interval, where the interior points lie. The theorem uses one more key condition. The function must have the same value at both endpoints. When that happens, Rolle’s Theorem guarantees at least one point...

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first-derivatives-and-the-shape-of-a-graph

Using First Derivatives to Find Graph Changes

The first derivative shows how a function changes across its domain. In calculus, the derivative is written as f'(x). It tells us whether the graph is rising, falling, or turning at a given point. A function is increasing on an interval when its derivative is positive. That means the slope of the tangent line is positive. A function is decreasing on an interval when its derivative is negative. In that case, the tangent line slopes downward. These sign changes help describe the overall shape...

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The first derivative test is used to find turning points on a smooth curve. It shows where a function changes from increasing to decreasing, or from decreasing to increasing. In this example, the curve can model an asset price that falls to a low point, rebounds, and then gradually declines again. The process starts by finding the first derivative of the function. Because the function is a product of a polynomial term and an exponential term, the product rule is needed. After differentiating,...

second derivatives and the shape of a graph

second-derivatives-and-the-shape-of-a-graph

Using Second Derivatives to Read Curvature

The second derivative shows how a graph bends. It gives key information about curvature and how that curvature changes over an interval. These ideas help with graph analysis in math and in real-world settings such as road elevation, population growth, and economic trends. A function f(x) is concave upward on an interval when its graph lies above all of its tangent lines. In this case, the second derivative is positive. That means the slope of the tangent line is increasing, and the graph bends...

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Classifying Critical Points with the Second Derivative

The second derivative test helps classify critical points of a function. Critical points are places where the first derivative is zero or undefined. They are possible locations for local maxima and local minima, but they do not always become one or the other. The second derivative shows concavity, or whether a graph bends up or down. If f''(x) &gt; 0, the function is concave up, and the critical point is a local minimum. If f''(x) &lt; 0, the function is concave down, and the critical point is...

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Using Derivatives to Read a Graph

Derivatives help students read the graph of a function by showing where the graph rises, falls, and bends. The first derivative tells the slope of the tangent line at each point. The second derivative shows how the curve changes shape over an interval. Points where the first derivative is zero or undefined are critical points. These points are important because they may mark a local maximum, a local minimum, or no extremum at all. A local minimum appears when the first derivative changes from...

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application-of-differentiation-to-business

Maximizing Bakery Revenue with Calculus

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Antiderivative Graph Shape and Slope

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application-of-antiderivatives-linear-motion

Finding Car Stopping Distance with Integration

Integration connects acceleration, velocity, and displacement in linear motion. It works as the reverse of a derivative, which tells how one quantity changes with another. In physics, this makes it possible to move from a rate of change back to the motion itself. A car provides a clear example. It is moving at a steady 20 meters per second when an obstacle appears 800 meters ahead. To stop in time, the car slows with a constant acceleration. That acceleration tells how the car’s velocity...

key terms and concepts

Learn how to find absolute maximum and minimum values on a closed interval by checking critical numbers, endpoints, and the Closed Interval Method.

Finding Absolute Extrema on a Closed Interval | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Absolute Extrema on a Closed Interval

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